5. Atoms and the periodic table of chemical elements. Definition of the geometrical structure of a molecule

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1 Historical introduction The Schrödinger equation for one-particle problems Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical elements 6 Diatomic molecules 7 Ten-electron systems from the second row 8 More complicated molecules FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Definition of the geometrical structure of a molecule Methane CH 4 Ammonia NH Water H O Dimethyl ether C H 6 O Ethyne C H Ethene C H 4 Ethane C H 6 Ethanol C H 6 O FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

2 The definition of molecular structure requires knowledge about type (ie charge number Z) and position (x, y, z) of the nuclei Only the relative position of the nuclei is important (translation and rigid rotation are excluded) Two frequently used systems of coordinates are (i) cartesian coordinates, and (ii) Z matrix coordinates The Z matrix definition uses a (suitably chosen) sequence of spherical coordinate systems, and gives values for radius r, polar distance θ and azimuthal angle ϕ A position vector R K, pointing from the origin of the coordinate system to the atomic nucleus, is thus known for each nucleus K FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Coordinates for ethanol, C H 5 OH: Cartesian coordinates: N data for an N-atomic molecule C C H H H H H O H Z matrix: N 6 data for a non-linear N-atomic molecule zmat angstroms c c cc h hc hcc h hc4 hcc4 dih4 h hc5 hcc5 4 dih5 h hc6 hcc6 dih6 h hc7 hcc7 4 dih7 o oc8 occ8 dih8 h 8 ho9 hoc9 dih9 variables cc hc hcc 0947 constants oc end FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

3 In our typical representations for molecules (crystals, etc) only the nuclear coordinates have some significance (we ignore, for the moment, the nuclear motion) Everything else, like balls, sticks, ribbons, etc is merely an eye-guide, though a very useful one FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Operators Operators Ô represent mathematical instructions, or operations These operations can be applied to suitably chosen mathematical objects, to yield new mathematical objects of the same or different kind Operators can be divided into classes, depending on the number of objects required for their application: Unary operators require a single mathematical object: Examples: Ô(a) = z f in f(x), eg f = ( ), f = sin (), f = exp () Important unary operators are linear operators L: L(x + y) = Lx + Ly FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

4 - L = d dx in f (x) = df dx - L = dx in dx f(x) = f(x) dx - the matrices in matrix-vector products: A (n m) x (m ) = b (n ) ( x T ( m) B (m k) = ct ( k) ( x T A ) ) x B ( = = ( b ) c T ) Binary operators require an ordered pair of objects: Ô(a, b) = z Examples: FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- + and in a + b and a b = a b scalar product and vector product in R : a b = a b cos (ϑ) = b a a = a, b = b, ϑ = (a, b) a b = a b sin (ϑ) n = b a n a = 0, n b = 0, n = matrix multiplication: A (n m) B (m k) = A (n m) B (m k) = C (n k) A B = C Ternary operators require an ordered triple of objects: Examples: Ô(a, b, c) = z FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

5 triple scalar product in R : a (b c) = a a a b b b c c c triple vector product in R : a (b c) = λ b µ c, λ = a c, µ = a b FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Groups Given one set of elements S = {a, b, } and one binary operation A group G = (S, ) is formed, if the following axioms hold: closure: a b = c with a, b, c G existence of a neutral element e: e a = a e = a existence of inverse elements a : a a = a a = e 4 associative law: a (b c) = (a b) c Subgroup: A subset of group elements, which constitutes a group (according to the criteria given above) Order of the group: The number of elements in the group is known as the order g = G of the group (g N, or g = for continuous groups) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

6 Abelian groups: If, in addition to the criteria given above, the commutative law a b = b a holds (ie all elements of the group commute), the group is called a commutative (or Abelian ) group Some well-known examples for Abelian groups: integers : (Z, +) real numbers : (R, +), (R \ {0}, ) (the order g is denumerably infinite in the first case, the last two cases represent continuous groups) Generators of a finite group: A subset of group elements from which all group elements can be formed (usually, there are several possible choices for a set of generators) N H Abel (80-89) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Permutation groups The permutations of n objects form a group The rule of combination (binary operation) is subsequent application This group is the permutation group, or symmetric group S n, which has order g = n! When the objects are simply the first n positive integers, a general notation for a permutation is ( ) 4 n P k =, i i i i i 4 i j i l, k n! (74) n The neutral element in a permutation group is the identity permutation, which leaves all n objects at their places: ( ) 4 n P = = e (75) 4 n FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

7 The next-to-trivial permutations are the permutations of pairs, or transpositions, eg ( ) 4 n P = (76) 4 n Another permutation, which involves already three objects, is ( ) 4 n P = 4 n (77) A shorter notation for the permutations is the notation with so-called cycles This gives for our examples ( ) 4 n P = = ()()()(4) (n) = e (78) 4 n ( ) 4 n P = = ()()(4) (n) (79) 4 n ( ) 4 n P = = ()(4) (n) (80) 4 n Cycles of length are usually omitted FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- A subset of the transpositions (ij) can be taken as generators for the permutation groups, ie all permutations can be expressed by a sequence of transpositions Depending on the number of transpositions involved, the permutations can be separated in even and odd permutations The former require an even number of transpositions, the latter require an odd number Some examples for n = : ( ) e = = ()()() 0 (even) ( ) = () (odd) ( ) = () = ()() (even) ( ) = () = ()() (even) Note: Start at the rightmost cycle, and work from right to left! FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

8 Permutations and functions Given two functions, depending on the position coordinates of two particles: Ψ a (, ) = f(r )g(r ) g(r )f(r ) (8) Ψ s (, ) = f(r )g(r ) + g(r )f(r ) (8) How are these functions affected by the permutation P = (), ie the interchange of the particles (or particle coordinates)? ()Ψ a (, ) = Ψ a (, ) = f(r )g(r ) g(r )f(r ) = Ψ a (, ) (8) ()Ψ s (, ) = Ψ s (, ) = f(r )g(r ) + g(r )f(r ) = + Ψ s (, ) (84) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Thus, Ψ a (, ) changes sign, or in other words, it is antisymmetric under this operation transposition of and, whereas Ψ s (, ) is not changed, in other words, it is symmetric under this operation The function Ψ a (, ) can be written in the form of a determinant: f(r Ψ a (, ) = ) f(r ) g(r ) g(r ) = f(r ) g(r ) (85) f(r ) g(r ) Such a determinant representation is always possible for a totally antisymmetric many-particle wave function, if it is approximated by products of single-particle functions Real- or complex-valued single-particle functions, which are square integrable, ie f (r) f(r) dr = N <, are called orbitals Singleparticle functions which include the spin coordinate, ie f(x) with x = (r, σ), are called spin orbitals A determinant built from spin orbitals is known as Slater determinant J C Slater ( ) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

9 A Slater determinant for three particles (n = ): φ (x ) φ (x ) φ (x ) Ψ(,, ) = N φ (x ) φ (x ) φ (x ) φ (x ) φ (x ) φ (x ) (86) N denotes a normalization constant The choice φ (x) = s(r)α(σ), φ (x) = s(r)β(σ), and φ (x) = s(r)α(σ) for the spin orbitals yields a valid approximate state function for the ground state of the Li atom, s s S Note that every particle uses every function, ie there is no relation or association between the coordinates (of a particle) and the singleparticle functions which have these coordinates as arguments A Li atom in the ground state has an occupied s orbital, but there is no s electron FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Symmetry groups (point groups, space groups) A set of symmetry operations (covering operations, which may be applied to a rigid body in -dimensional space, eg a molecule with fixed structure) constitutes a symmetry group The rule of combination (binary operation) is subsequent application the se- Symmetry groups are, in general, non-abelian groups, ie quence of application of symmetry operations is important At least one point remains fixed in space under all point group symmetry operations, while space groups include also translations as symmetry operations Operation (in connection with symmetry groups): A transformation of coordinates, or alternatively a transformation of a molecule to a new position FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

10 Symmetry operation: An operation (not necessarily a physically feasible one) that carries a molecule into a new position which is indistinguishable from (or equivalent to) the original position Proper operations: Pure rotations about a specified axis (these are physically feasible) Improper operations: These may be regarded as rotations-reflections (or alternatively rotations-inversions, these are not physically feasible) Symmetry element: A geometrical entity (point, line or plane) related to a symmetry operation FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Symmetry element Symmetry Definition of Symbol a Name operations symmetry operations Ê Identity (neutral element) Proper symmetry operations C n n-fold rotation axis Ĉn k Rotation through φ = k π/n, (usually assumed to k n, about the principal axis be in the z direction) C, C -fold rotation axis Ĉ, Ĉ Rotation through φ = π perpendicular to the about the axis principal C n axis Improper symmetry operations S n n-fold rotation- Ŝn k Rotation through φ = k π/n, reflection axis combined with reflection k times in a plane normal to the axis, n even: k n, n odd: k n i (= S ) inversion center î (= Ŝ ) Inversion through the origin σ (= S ) mirror plane σ (= Ŝ ) Reflection in a plane σ v, σ h, σ d (vertical, horizontal, σ v, σ h, σ d dihedral planes) a Notation due to A Schoenflies (85-98) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

11 The symmetry operation Ĉ and the water molecule φ = π/ φ = π/ y φ = π x FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Symmetry operations and the ammonia molecule φ = π/ σ σ σ φ=π/ y x σ FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

12 An algorithm to determine the point group from symmetry elements Linear? y n i? Unique C n of highest order? n y n y T C v D h 4 C? i? n σ d? y n n y n y n y n y n C 4? C? I I h n C C n? S 4? i? σ? σ h? n σ v? y n n y n y n y y n T d i? O O h i? 6 C 5? S n C n? n y n y C s D nh T h C C i D n D d C n C nh C nv σ d? σ h? n y n y n y n y S n D nd J A Salthouse, M J Ware: Point group character tables and related data Cambridge, 97 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Determination of point groups C nv (Examples: H O [n = ], NH [n = ]) Linear? y n i? Unique C n of highest order? n y n y T C v D h 4 C? i? n σ d? y n n y n y n y n y n C 4? C? I I h n C C n? S 4? i? σ? σ h? n σ v? y n n y n y n y y n T d i? O O h i? 6 C 5? S n C n? n y n y C s D nh T h C C i D n D d C n C nh C nv σ d? σ h? n y n y n y n y S n D nd J A Salthouse, M J Ware: Point group character tables and related data Cambridge, 97 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

13 T Determination of the point group D h (Example: C H 4 ) Linear? y n i? Unique C n of highest order? n y n y C v D h 4 C? i? n σ d? y n n y n y n y n y n C 4? C? I I h n C C n? S 4? i? σ? σ h? n σ v? y n n y n y n y y n T d i? O O h i? 6 C 5? S n C n? n y n y C s D h T h C C i D n D d C n C nh C nv σ d? σ h? n y n y n y n y S n D nd J A Salthouse, M J Ware: Point group character tables and related data Cambridge, 97 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Point group Generating Symmetry Order Comments symbol operations elements g C Ê none no symmetry C s σ σ C s = C h = C v = S C i î i C i = S C n Ĉ n C n n S n Ŝ n C n, S n n C nh Ĉ n, σ h C n, σ h, S n n if n odd: C nh = S n C nv Ĉ n, σ v C n, n σ v n n-gonal regular pyramid D n Ĉ n, Ĉ C n, n C n D nh Ĉ n, Ĉ, σ h C n, n C, S n, σ h, n σ v 4n n-gonal archimedian prism D nd Ĉ n, Ĉ, σ d C n, n C, n σ d, S n 4n n-gonal archimedian antiprism C v Ĉ (z), σ v C, σ v Ĉ (z), Ĉ, σ h C, σ v, S, C D h T T h T d O O h I I h Ĉ (xyz), Ĉ (z) Ĉ (xyz), Ĉ (z) Ĉ (xyz), Ŝ (z) Ĉ (xyz), Ĉ (z) Ĉ (xyz), Ĉ (z) Ĉ (ico), Ĉ (z) Ĉ (ico), Ĉ (z) 4 C, C, î 4 C, C, 4 S 6, σ v C, C, S 4, 6 σ d 4 regular tetrahedron 4 4 C, C 4, 6 C 4 4, î 4 C, C 4, 6 C, S 4, 48 4 S 6, σ h, 6 σ d regular octahedron 5 6 C 5, 0 C, 5 C 60 5, î 6 C 5, 0 C, 5 C, 0 S 0, 0 S 6, 5 σ regular icosahedron K Ĉ C K h Ĉ, î C, S sphere FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

14 Transformation of vectors in R Introduce a suitable basis to describe the (active) rotation of a position vector r around an axis through the origin in direction of the unit vector n by an angle φ into a new position vector r = R(φn) r: The component of r parallel to n: r = (n r) n = n (n r) x r z n r y The orthogonal complement to r : r = r r The vector normal to the plane spanned by n and r: n = n n n n = n r = n (r + r ) = n r FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- The new position vector r, which results from the action of the operation R(φn) on the old position vector r, is now representable as a linear combination of these basis vectors: r = R(φn) r = a r + b n r + c r = a r + b n r + (c a) (n r) n = (a + b n +(c a) n n ) r with a = cos (φ), b = sin (φ), and c = + for pure rotations (or c = for rotations-reflections) For the application of any symmetry operation R in a point group to a point r in space we may thus write r = R ± (φn) r, or r = R ± (φn) r in matrix-vector notation Explicitly, with coordinates (or vector components): x y z = a + (c a)n (c a)n n bn (c a)n n + bn (c a)n n + bn a + (c a)n (c a)n n bn (c a)n n bn (c a)n n + bn a + (c a)n x y z (87) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

15 With φ = π/n and suitably chosen n: R = Ĉn R + (φn), R = Ŝn R (φn) (88) For every point group, the resulting set of matrices R ± (φn) forms a group under matrix multiplication, which is isomorphic to the point group Transformation of scalar functions With knowledge about the tranformation of position vectors, r, the transformation law for scalar functions of the coordinates, f(r), can be derived The condition of equality of function values, ie the transformed function ÔRf shall have at the transformed position r = Rr the same function value as the original function f at the original position r = R r, leads to: Ô R f(r ) = f(r) = f( R r ) ÔRf(r) = f( R r) (89) since this relation should be valid for every argument r in R FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- What happens to the function f(r) = f(x, y, z) = x y exp ( r ) under the counterclockwise rotation around the z axis (n = (0, 0, ) T ) through an angle φ = π/4 = π/8? Ĉ 8 R = Ĉ 8 R = R r = R x y z = (x y) (x + y) Ô R f(r) = f(r r) = f( (x y), (x + y), z) = (x y ) exp ( r ) z 0 y x z = 0 0 FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

16 Fields Given one set of elements S = {a, b, } and two binary operations and A field F = (S,, ) is formed, if the following axioms hold: (S, ) is an Abelian group (with neutral element 0) (S \ {0}, ) is an Abelian group (with neutral element ) distributive laws: a (b c) = (a b) (a c), (a b) c = (a c) (b c) Some well-known examples: rational numbers : (Q, +, ) real numbers : (R, +, ) complex numbers : (C, +, ) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- Linear vector spaces Given two sets (in fact not only sets, but algebraic structures): () an abelian group G = (S, ), generated from a set of vectors S = { a, b,, v, }, and a binary operation called vector addition (with the null vector o as neutral element), () a field F = (K, +, ) (usually K = R or K = C) with elements α, β, γ,, called scalars A linear vector space V over the field F is formed, if in addition to the above a scalar multiplication (S multiplication) is defined as: distributive and associative laws hold: α a = a α = v V (90) (α + β) a = α a β a α ( a b ) = α a α b (9) (α β) a = α (β a ) These are called ket vectors in the Dirac notation used here FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

17 Usually, the binary operations + and (addition of scalars and vectors, respectively) are not distinguished any further, and only the symbol + is used to denote both For K = R the linear vector space V is called a real linear vector space, whereas for K = C it is called a complex linear vector space These definitions include already, or are easily extended to include: (a) linear combinations, ie a weighted sum of an arbitrary finite number l of vectors (the limit l requires further study): l v = α i a i = i= If, for the special case v = o, l i= a i α i (9) l α i a i = o α i = 0 (for all i) (9) i= FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- ie there exists only the trivial solution, then the set of vectors { a i } is called linearly independent (and linearly dependent otherwise) (b) a basis (or basis set), ie a set of vectors { b k } which is linearly independent and capable of representing an arbitrary vector v through linear combination: n k= β k b k = v for any v V (94) The number n (n N) of basis vectors is the dimension of V This dimension can be finite (n < ) or denumerably infinite (n = ), and even the case of a continuum (n = ) could be included, if we change the discrete summation in eq (94) to an integration in a suitable way For n =, however, the convergence of the expansion in eq (94) cannot be taken for granted If convergence (pointwise or in the mean) holds for n =, the set { b k } is called complete FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

18 With any ordered pair of vectors, ( u, v ), may be associated a scalar product u v K A scalar product has, in general, the following properties: - v v R; v v 0; v v = 0 v = o ; - u v = v u ; - α u + α u v = α u v + α u v and u β v + β v = β u v + β u v A linear vector space with scalar product is also known as inner product space or pre-hilbert space A scalar product can be used to define the length (or norm) of a vector: v = v v 0 (95) in which case the linear vector space turns into a unitary space, where the following relations hold: This is called a bra-c-ket in the Dirac notation used here D Hilbert (86-94) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- - triangle inequality: u + v ( u + v ) ; - Cauchy-Schwarz inequality: u v = u v u v = u v v u u u v v ; - parallelogram equality: u + v + u v = ( u + v ) A vector v with v = is called normalized to unity (or just normalized ) Two vectors u, v with u v = 0 are called orthogonal to each other (or just orthogonal ) A set of vectors { v k } (k =,, m) with { for k = l v k v l = δ kl = 0 for k l is called an orthonormal set (96) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

19 An orthonormal set of basis vectors, { b k } (k =,, n) with b k b l = δ kl, makes the evaluation of the expansion coefficients β k in a linear combination particularly simple: b k b l = δ kl v = n k= b k β k, β k = b k v (97) Substitution of this expression for β k gives an expression for the identity (or unit) operator which is known as resolution of the identity : n n v = b k b k v = b k b k v = v (98) k= = k= n k= b k b k (99) The distance d between two vectors u, v can be obtained from d = u v = v u, (00) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- and with restriction to cases where u v R the angle φ = ( u, v ) between them can be defined as cos (φ) = u v u v (0 φ π) (0) Hilbert space: A complete unitary linear vector space (a rigorous definition is not attempted here) Some examples for linear spaces: v v (associate ket vectors with ordinary vectors): This yields the n-dimensional Euclidean space R n with a basis set {b k } (k =,, n), so that the expansions c = n k= b k γ k = (b,, b n ) γ γ n, d = exist and a scalar product can be defined as n k= b k δ k FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

20 c d = c T Md, with c T = (γ,, γ n ) and d = δ, δ n and the metric matrix M = (m ij ) where m ij = b i b j = m ji When the basis is chosen to be orthonormal (b i b j = δ ij ), the metric matrix reduces to the n n unit matrix, and the scalar product collapses to the simple familiar form where only the coefficients (coordinates) γ k and δ k are involved: b i b j = δ ij c d = c T d = (γ,, γ n ) δ δ n = n k= γ k δ k f f (associate ket vectors with ordinary functions): This leads, eg, to the infinite-dimensional spaces L (G) of complex-valued functions of n variables, f(r) = f(x, x,, x n ) C (r R n ), that are square-integrable over a range (or region) G R n in the sense of the scalar product f g = f (r) g(r) dr G FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- This leads to nor- based on the Lebesgue integral definition malization integrals ) / = ( ( ) / f = f (r) f(r) dr G G f(r) dr Special cases included herein are - L ([ p/, p/]) used in the harmonic analysis of periodic functions f(x) = f(x + p) C; for period p = π: Orthonormal basis (dimension d =, denumerably infinite): { } e ikx π (k Z), π π π e i(k l)x dx = δ kl Fourier series expansion: f(x) = k= c k e ikx, c k = π π π π e ikx f(x) dx H Lebesgue (875-94) J Fourier (768-80) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

21 - L (R) and Fourier transformation: Orthonormal basis (dimension d =, continuum): { } e ikx (k R), π π e i(k k )x dx = δ(k k ) Fourier transform pairs (symmetric form): f(x) = f(k) π eikx dk, f(k) = π f(x) e ikx dx - L (R ) and the construction of molecular orbitals ψ k (r) by linear combination of basis functions χ i (r) (for historical reasons, this is called molecular orbitals by linear combination of atomic orbitals, or MO-LCAO approach ): or equivalently ψ k (r) = m i= χ i (r) c ik (ψ, ψ,, ψ m ) = (χ, χ,, χ m ) C FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4- when the MO coefficients c ik are collected into MO (column) vectors c k, which form the matrix C Completeness of the basis set is a delicate issue, which is usually left untouched in practical work - L ((R S) n ) [ where S denotes a spin space ] and the construction of (spin-adapted) n-electron state functions Ψ k by linear combination of Slater determinants Φ i (called configuration interaction or CI expansion, since every Slater determinant can be associated with an electron configuration ): or equivalently Ψ k (x,, x n ) = m i= Φ i (x,, x n ) C ik (Ψ, Ψ,, Ψ m ) = (Φ, Φ,, Φ m ) C when the CI coefficients C ik are collected into CI (column) vectors C k, which form the matrix C In the limit m, the CI expansion is exact, if the spin orbitals from which the Slater determinants are formed constitute a complete basis (of single-particle functions) FAQC D Andrae, Theoretical Chemistry, U Bielefeld / 4-

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